Theorem suppimacnv 7306

Description: Support sets of functions expressed by inverse images. (Contributed by AV, 31-Mar-2019.) (Revised by AV, 7-Apr-2019.)

Assertion

Ref	Expression
suppimacnv	|- ( ( R e. V /\ Z e. W ) -> ( R supp Z ) = ( `' R " ( _V \ { Z } ) ) )

Proof of Theorem suppimacnv

Dummy variables x y s t are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq2 4657	. . . . . . . 8 |- ( t = s -> ( x R t <-> x R s ) )
2	1	cbvexvw 1970	. . . . . . 7 |- ( E. t x R t <-> E. s x R s )
3		breq2 4657	. . . . . . . . . . . . . 14 |- ( s = Z -> ( x R s <-> x R Z ) )
4	3	anbi1d 741	. . . . . . . . . . . . 13 |- ( s = Z -> ( ( x R s /\ ( x R t <-> t =/= Z ) ) <-> ( x R Z /\ ( x R t <-> t =/= Z ) ) ) )
5		bianir 1009	. . . . . . . . . . . . . . . . . 18 |- ( ( t =/= Z /\ ( x R t <-> t =/= Z ) ) -> x R t )
6		vex 3203	. . . . . . . . . . . . . . . . . . . 20 |- t e. _V
7		breq2 4657	. . . . . . . . . . . . . . . . . . . . 21 |- ( y = t -> ( x R y <-> x R t ) )
8		neeq1 2856	. . . . . . . . . . . . . . . . . . . . 21 |- ( y = t -> ( y =/= Z <-> t =/= Z ) )
9	7, 8	anbi12d 747	. . . . . . . . . . . . . . . . . . . 20 |- ( y = t -> ( ( x R y /\ y =/= Z ) <-> ( x R t /\ t =/= Z ) ) )
10	6, 9	spcev 3300	. . . . . . . . . . . . . . . . . . 19 |- ( ( x R t /\ t =/= Z ) -> E. y ( x R y /\ y =/= Z ) )
11	10	ex 450	. . . . . . . . . . . . . . . . . 18 |- ( x R t -> ( t =/= Z -> E. y ( x R y /\ y =/= Z ) ) )
12	5, 11	syl 17	. . . . . . . . . . . . . . . . 17 |- ( ( t =/= Z /\ ( x R t <-> t =/= Z ) ) -> ( t =/= Z -> E. y ( x R y /\ y =/= Z ) ) )
13	12	ex 450	. . . . . . . . . . . . . . . 16 |- ( t =/= Z -> ( ( x R t <-> t =/= Z ) -> ( t =/= Z -> E. y ( x R y /\ y =/= Z ) ) ) )
14	13	pm2.43a 54	. . . . . . . . . . . . . . 15 |- ( t =/= Z -> ( ( x R t <-> t =/= Z ) -> E. y ( x R y /\ y =/= Z ) ) )
15	14	adantld 483	. . . . . . . . . . . . . 14 |- ( t =/= Z -> ( ( x R Z /\ ( x R t <-> t =/= Z ) ) -> E. y ( x R y /\ y =/= Z ) ) )
16		nne 2798	. . . . . . . . . . . . . . . 16 |- ( -. t =/= Z <-> t = Z )
17		notbi 309	. . . . . . . . . . . . . . . . . . . 20 |- ( ( x R t <-> t =/= Z ) <-> ( -. x R t <-> -. t =/= Z ) )
18		bianir 1009	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( -. t =/= Z /\ ( -. x R t <-> -. t =/= Z ) ) -> -. x R t )
19		breq2 4657	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( Z = t -> ( x R Z <-> x R t ) )
20	19	eqcoms 2630	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( t = Z -> ( x R Z <-> x R t ) )
21		pm2.24 121	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( x R t -> ( -. x R t -> E. y ( x R y /\ y =/= Z ) ) )
22	20, 21	syl6bi 243	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( t = Z -> ( x R Z -> ( -. x R t -> E. y ( x R y /\ y =/= Z ) ) ) )
23	22	com13 88	. . . . . . . . . . . . . . . . . . . . . 22 |- ( -. x R t -> ( x R Z -> ( t = Z -> E. y ( x R y /\ y =/= Z ) ) ) )
24	18, 23	syl 17	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( -. t =/= Z /\ ( -. x R t <-> -. t =/= Z ) ) -> ( x R Z -> ( t = Z -> E. y ( x R y /\ y =/= Z ) ) ) )
25	24	ex 450	. . . . . . . . . . . . . . . . . . . 20 |- ( -. t =/= Z -> ( ( -. x R t <-> -. t =/= Z ) -> ( x R Z -> ( t = Z -> E. y ( x R y /\ y =/= Z ) ) ) ) )
26	17, 25	syl5bi 232	. . . . . . . . . . . . . . . . . . 19 |- ( -. t =/= Z -> ( ( x R t <-> t =/= Z ) -> ( x R Z -> ( t = Z -> E. y ( x R y /\ y =/= Z ) ) ) ) )
27	26	com13 88	. . . . . . . . . . . . . . . . . 18 |- ( x R Z -> ( ( x R t <-> t =/= Z ) -> ( -. t =/= Z -> ( t = Z -> E. y ( x R y /\ y =/= Z ) ) ) ) )
28	27	imp 445	. . . . . . . . . . . . . . . . 17 |- ( ( x R Z /\ ( x R t <-> t =/= Z ) ) -> ( -. t =/= Z -> ( t = Z -> E. y ( x R y /\ y =/= Z ) ) ) )
29	28	com13 88	. . . . . . . . . . . . . . . 16 |- ( t = Z -> ( -. t =/= Z -> ( ( x R Z /\ ( x R t <-> t =/= Z ) ) -> E. y ( x R y /\ y =/= Z ) ) ) )
30	16, 29	sylbi 207	. . . . . . . . . . . . . . 15 |- ( -. t =/= Z -> ( -. t =/= Z -> ( ( x R Z /\ ( x R t <-> t =/= Z ) ) -> E. y ( x R y /\ y =/= Z ) ) ) )
31	30	pm2.43i 52	. . . . . . . . . . . . . 14 |- ( -. t =/= Z -> ( ( x R Z /\ ( x R t <-> t =/= Z ) ) -> E. y ( x R y /\ y =/= Z ) ) )
32	15, 31	pm2.61i 176	. . . . . . . . . . . . 13 |- ( ( x R Z /\ ( x R t <-> t =/= Z ) ) -> E. y ( x R y /\ y =/= Z ) )
33	4, 32	syl6bi 243	. . . . . . . . . . . 12 |- ( s = Z -> ( ( x R s /\ ( x R t <-> t =/= Z ) ) -> E. y ( x R y /\ y =/= Z ) ) )
34		vex 3203	. . . . . . . . . . . . . . . 16 |- s e. _V
35		breq2 4657	. . . . . . . . . . . . . . . . 17 |- ( y = s -> ( x R y <-> x R s ) )
36		neeq1 2856	. . . . . . . . . . . . . . . . 17 |- ( y = s -> ( y =/= Z <-> s =/= Z ) )
37	35, 36	anbi12d 747	. . . . . . . . . . . . . . . 16 |- ( y = s -> ( ( x R y /\ y =/= Z ) <-> ( x R s /\ s =/= Z ) ) )
38	34, 37	spcev 3300	. . . . . . . . . . . . . . 15 |- ( ( x R s /\ s =/= Z ) -> E. y ( x R y /\ y =/= Z ) )
39	38	ex 450	. . . . . . . . . . . . . 14 |- ( x R s -> ( s =/= Z -> E. y ( x R y /\ y =/= Z ) ) )
40	39	adantr 481	. . . . . . . . . . . . 13 |- ( ( x R s /\ ( x R t <-> t =/= Z ) ) -> ( s =/= Z -> E. y ( x R y /\ y =/= Z ) ) )
41	40	com12 32	. . . . . . . . . . . 12 |- ( s =/= Z -> ( ( x R s /\ ( x R t <-> t =/= Z ) ) -> E. y ( x R y /\ y =/= Z ) ) )
42	33, 41	pm2.61ine 2877	. . . . . . . . . . 11 |- ( ( x R s /\ ( x R t <-> t =/= Z ) ) -> E. y ( x R y /\ y =/= Z ) )
43	42	expcom 451	. . . . . . . . . 10 |- ( ( x R t <-> t =/= Z ) -> ( x R s -> E. y ( x R y /\ y =/= Z ) ) )
44	43	exlimiv 1858	. . . . . . . . 9 |- ( E. t ( x R t <-> t =/= Z ) -> ( x R s -> E. y ( x R y /\ y =/= Z ) ) )
45	44	com12 32	. . . . . . . 8 |- ( x R s -> ( E. t ( x R t <-> t =/= Z ) -> E. y ( x R y /\ y =/= Z ) ) )
46	45	exlimiv 1858	. . . . . . 7 |- ( E. s x R s -> ( E. t ( x R t <-> t =/= Z ) -> E. y ( x R y /\ y =/= Z ) ) )
47	2, 46	sylbi 207	. . . . . 6 |- ( E. t x R t -> ( E. t ( x R t <-> t =/= Z ) -> E. y ( x R y /\ y =/= Z ) ) )
48	47	imp 445	. . . . 5 |- ( ( E. t x R t /\ E. t ( x R t <-> t =/= Z ) ) -> E. y ( x R y /\ y =/= Z ) )
49	48	a1i 11	. . . 4 |- ( ( R e. V /\ Z e. W ) -> ( ( E. t x R t /\ E. t ( x R t <-> t =/= Z ) ) -> E. y ( x R y /\ y =/= Z ) ) )
50	49	ss2abdv 3675	. . 3 |- ( ( R e. V /\ Z e. W ) -> { x | ( E. t x R t /\ E. t ( x R t <-> t =/= Z ) ) } C_ { x | E. y ( x R y /\ y =/= Z ) } )
51		suppvalbr 7299	. . 3 |- ( ( R e. V /\ Z e. W ) -> ( R supp Z ) = { x | ( E. t x R t /\ E. t ( x R t <-> t =/= Z ) ) } )
52		cnvimadfsn 7304	. . . 4 |- ( `' R " ( _V \ { Z } ) ) = { x | E. y ( x R y /\ y =/= Z ) }
53	52	a1i 11	. . 3 |- ( ( R e. V /\ Z e. W ) -> ( `' R " ( _V \ { Z } ) ) = { x | E. y ( x R y /\ y =/= Z ) } )
54	50, 51, 53	3sstr4d 3648	. 2 |- ( ( R e. V /\ Z e. W ) -> ( R supp Z ) C_ ( `' R " ( _V \ { Z } ) ) )
55		suppimacnvss 7305	. 2 |- ( ( R e. V /\ Z e. W ) -> ( `' R " ( _V \ { Z } ) ) C_ ( R supp Z ) )
56	54, 55	eqssd 3620	1 |- ( ( R e. V /\ Z e. W ) -> ( R supp Z ) = ( `' R " ( _V \ { Z } ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   =/= wne 2794   _Vcvv 3200   \ cdif 3571   {csn 4177   class class class wbr 4653   `'ccnv 5113   "cima 5117  (class class class)co 6650   supp csupp 7295

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-supp 7296

This theorem is referenced by:  frnsuppeq  7307  suppun  7315  mptsuppdifd  7317  supp0cosupp0  7334  imacosupp  7335  fdmfisuppfi  8284  fsuppun  8294  fsuppco  8307  gsumval3a  18304  gsumzf1o  18313  gsumzaddlem  18321  gsumzmhm  18337  gsumzoppg  18344  deg1val  23856  suppss3  29502  ffsrn  29504  fpwrelmapffslem  29507  sitgclg  30404  eulerpartlemmf  30437  eulerpartlemgf  30441  fidmfisupp  39390